Problem 47 - Beginning and Intermediate Algebra

Question

A company's advertising budget is currently \(\$ 500,000\) per year. For the next several years, they will cut the budget by \(10 \%\) per year. a) Find the general term, \(a_{m}\) of the geometric sequence that models the company's advertising budget for each of the next several years. b) What is the advertising budget 3 yr from now?

Step-by-Step Solution

Answer

a) \(a_{m} = 500,000 * 0.9^{(m-1)}\) b) \$364,500

Step 1: Identify Initial Term and Common Ratio

The initial term of the geometric sequence is the current advertising budget, which is \(a_1 = \$500,000\). The common ratio, r, is the percentage by which the company's advertising budget will be reduced each year. In this case, the budget is cut by \(10\%\). So the common ratio is r = \(1 - 0.1 = 0.9\).

Step 2: Find the General Term

Now that we have the initial term and the common ratio, we can find the general term \(a_{m}\) for the geometric sequence. The formula for the mth term of a geometric sequence is \(a_{m} = a_1 * r^(m-1)\). In this case, we have \(a_{m} = 500,000 * 0.9^{(m-1)}\).

Step 3: Calculate Advertising Budget 3 Years from Now

We need to find the advertising budget 3 years from now, which means we want to find the value of the sequence at m = 4 (since the first year is the current year). Using the formula for the general term, we'll plug in m = 4 in the formula: \(a_{4} = 500,000 * 0.9^{(4-1)}\) \(a_{4} = 500,000 * 0.9^{3}\) \(a_{4} = 500,000 * 0.729\) \(a_{4} = \$364,500\) So the advertising budget 3 years from now will be \$364,500. In summary: a) The general term for this geometric sequence is \(a_{m} = 500,000 * 0.9^{(m-1)}\) b) The advertising budget 3 years from now is \$364,500.

Key Concepts

Advertising Budget

When we talk about an advertising budget, we're referring to the amount of money a company sets aside for its marketing campaigns over a specific period. This budget is crucial because it dictates how much a company can invest in promoting its products and services.
In the context of our exercise, the company has an advertising budget starting at \(\$500,000\) per year. However, due to business strategies, they decide to decrease it annually by a percentage. Changes like these—where the budget is systematically cut or increased—can often be modeled using geometric sequences.
Understanding your advertising budget helps in crafting more effective marketing plans and ensures sustainable financial planning.

Common Ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It describes how the sequence progresses.
In our scenario, the company's budget is reduced by \(10\%\) each year. This means that the new budget each year is \(90\%\) of what it was the previous year. We determine the common ratio \(r\) by subtracting the percentage decrease from 1:

• The common ratio \(r = 1 - 0.1 = 0.9\).

The common ratio indicates that each year's budget is \(\text{90\%}\) of the previous year's budget, establishing a predictable pattern for planning future spending.

Initial Term

The initial term of a geometric sequence is essentially its starting point. In business terms, it is your starting value before any changes take place.
In this problem, the initial term of the sequence, \(a_1\), is the current advertising budget, which stands at \(\$500,000\). Understanding this initial term is crucial because it serves as the baseline.
This baseline will be affected by the changes due to the common ratio over time, showing how the advertising budget evolves with each sequential year.

General Term

The general term of a geometric sequence enables us to calculate the value of any term in the sequence without listing all previous terms. It's a mathematical tool that simplifies predictions and planning.
The formula for the \(m\)th term in a geometric sequence is given by:

• \(a_{m} = a_1 \cdot r^{(m-1)}\)

Using this formula, one can compute forecasts like the future value of the advertising budget. For example, to find out what the advertising budget will be three years from now, plug \(m = 4\) into the general formula:

• \(a_{4} = 500,000 \cdot 0.9^{3}\)
• Calculations reveal that \(a_{4} = \$364,500\), reflecting the budget adjustment for planning purposes.

Having a clear understanding of the general term helps with strategic decision-making and long-term financial strategies.